Let's consider a martingale $X_n$ where $E(X^2_n)<\infty$ and define $D_i=X_i-X_{i-1}$, $i\geq 2$. Show that
(a) $E[D_i]=0$
(b) $cov(D_i,D_j)=0$, for each $i \neq j$;
and conclude that
(c) $Var(X_n)=Var(x_1)+\displaystyle\sum_{i=2}^{n}Var(D_i)$.
I usually know how to proof easy martingales but I really don't know where to start this one. Thank You
For the first point, you can condition on $\mathcal{F_{i-1}}$ (to which $X_i$ is adapted) as GNU Supporter mentions in the hint:
$$\mathbb{E}[D_i] = \mathbb{E}[X_i - X_{i-1}] \underbrace{=}_{\text{tower property}} \mathbb{E}[\mathbb{E}[X_i - X_{i-1} \vert \mathcal{F_{i-1}}]] = \mathbb{E}[X_i \vert \mathcal{F_{i-1}}] - \mathbb{E}[X_{i-1}] = 0$$
For the second point, since $\mathbb{E}[\vert X_i \vert^2] \lt \infty$ you have $\mathbb{E}[\vert D_i \vert^2] \lt \infty$. Assume $i \leq j$ then by definition of the covariance
$$\text{Cov}(D_i,D_j) = \mathbb{E}[(D_i - \mathbb{E}[D_i])(D_j - \mathbb{E}[D_j])] = \mathbb{E}[D_i D_j] \\ = \mathbb{E}[\mathbb{E}[D_i D_j \vert F_{j-1}]] = \mathbb{E}[D_i \underbrace{\mathbb{E}[D_j \vert F_{j-1}]}_{= 0}] = 0 $$
For the last point, write $X_n$ in terms of $X_1$ and $D_i$ ($i \in \{2,\dots,n\}$) and use the identity that holds for pairwise independent variables (so, you still need to prove $X_1 \perp D_i$ for $i \in \{2,\dots,n\}$)
$$\text{Var}(\sum_{i=1}^n A_i) = \sum_{i=1}^n \text{Var}(A_i)$$